Let [ t ] denote the greatest integer ≤ t . Then (2)/(π) ∫ _...

Let,

$I = {∫}_{\dfrac{\pi}{6}}^{\dfrac{5\pi}{6}}\left(8\left[cosecx\right] - 5\left[\cot x\right]\right)dx$

$⇒ I = 8{∫}_{\dfrac{\pi}{6}}^{\dfrac{5\pi}{6}}\left[cosecx\right]dx - 5{∫}_{\dfrac{\pi}{6}}^{\dfrac{5\pi}{6}}\left[\cot x\right]dx$

$⇒ I = 8{I}_{1} - 5{I}_{2}$, where ${I}_{1} = 8{∫}_{\dfrac{\pi}{6}}^{\dfrac{5\pi}{6}}\left[cosecx\right]dx \text{\text{ and } } {I}_{2} = 5{∫}_{\dfrac{\pi}{6}}^{\dfrac{5\pi}{6}}\left[\cot x\right]dx$

Now solving,

${I}_{1} = {∫}_{\dfrac{\pi}{6}}^{\dfrac{5\pi}{6}}\left[cosecx\right]dx$

$⇒ {I}_{1} = {∫}_{\dfrac{\pi}{6}}^{\dfrac{5\pi}{6}}1dx = \dfrac{2\pi}{3}$

As when $x \in \left(\dfrac{\pi}{6}, \dfrac{5\pi}{6}\right), cosecx \in [ 1, 2 )$,

So, \left[cosecx\right] = 1$</p><p>Now solving, </p><p>${I}_{2} = {∫}_{\dfrac{\pi}{6}}^{\dfrac{5\pi}{6}}\left[\cot x\right]dx$</p><p>$⇒ {I}_{2} = {∫}_{\dfrac{\pi}{6}}^{\dfrac{\pi}{4}}1dx + {∫}_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}}0dx + {∫}_{\dfrac{\pi}{2}}^{\dfrac{3\pi}{4}}\left(- 1\right)dx + {∫}_{\dfrac{3\pi}{4}}^{\dfrac{5\pi}{6}}\left(- 2\right)dx$</p><p>$⇒ {I}_{2} = \left(\dfrac{\pi}{4} - \dfrac{\pi}{6}\right) - \left(\dfrac{3\pi}{4} - \dfrac{\pi}{2}\right) - 2\left(\dfrac{5\pi}{6} - \dfrac{3\pi}{4}\right)$</p><p>$⇒ {I}_{2} = - \dfrac{\pi}{3}$</p><p>Required value will be,</p><p>$\dfrac{2}{\pi}I = \dfrac{2}{\pi}\left[8 \times \dfrac{2\pi}{3} + 5 \times \dfrac{\pi}{3}\right] = 14